Group Therapy - The Problem: How Accurate Is Your Rifle?

If you think it does, or think it doesn’t, maybe you’ll find this look at group size will change your mind.

One fine day, I ran into my friend Jim at the range. Jim is a real gentleman, and he delights in fine-tuning his rifles. “Hey Jim, did you get that 308 of yours settled down and shooting?”

“Sure did. I’ve got it shooting 5/8-inch. Of course, I got it to do that only once.”

I smiled. My friend was in the midst of a common misunderstanding about testing group size. He thought that the 5/8" group was the one time he did everything right, that the other times, his technique wasn’t perfect. The illusion is that if his technique were better, he could consistently get that group size. But it just isn’t so. Rifles produce larger and smaller groups without an identifiable cause. The trick is knowing when a change is big enough to be attributed to an identifiable cause, and not the sum of the many, tiny, unidentifiable causes. More than likely Jim’s rifle was not displaying the results of an identifiable cause, and he hadn’t done anything really different to get that 5/8-inch group.

A second issue is that it makes no sense whatever to speak of group size without also specifying the number of shots. If you think about it for a moment, no group will ever be smaller than the spacing between the first two shots. Groups can only get bigger as you add more shots. So, for the same accuracy, and all other factors equal, groups with more shots will, on average, be larger than groups with fewer shots. One thing we need is the ability to convert back and forth between average group sizes shot with different numbers of shots.

In this article, I will attempt to address both issues, separating real change from normal random variation, and properly comparing average group size when different numbers of shots are fired for each group.

Some Mathematical Musings
Measuring the center-to-center distance between the two most distant bullet holes is the most widely accepted method for measuring group size. This is one measure I will use in this article.

Another tool I will use is standard deviation. That is a measure used both in and out of the shooting hobby, and I used it to create the simulation needed to find what constitutes “usual” and “unusual” behavior.

Such measures as mean distance from the center contain no information that is not in standard deviation, and they are much more difficult to work with. So let’s keep it simple, and use the widely accepted measures.

An Important Definition
The limiting factors for accuracy can be divided into two groups. One group is problems that can be identified and eliminated, such as a stock putting pressure on a barrel, shooter flinch, or a loose scope. These are called “special cause.” The other group is made up of many tiny causes lumped together, that are characteristic of the process, and probably always present. These might include normal tiny imbalances in the bullet, a speck of grit on the crown, and normal random variation in bullet weight. We lump these together as “common cause,” and treat them statistically.

In order for what follows to work properly, the rifle must be free from special cause, with only normal, random variation (common cause) remaining. In other words, you have to have already gotten the rifle settled down and shooting as well as it can.

A Terrible Choice: Imperfect Information, Or No Longer Interesting Information
One way to get a real truth about group size is to start with a very large pile of ammunition, and shoot groups of the same size until the barrel wears out. The average group size we get by measuring all the groups is the true average group size for that number of shots. We might think of this as being represented by Figure 1. Unfortunately, this information is no longer useful, because the barrel is now worn out, and of no further use. So we usually use a different approach.

Figure 1

That other approach is to shoot a small number of shots — a sample. This will not give a perfectly accurate result. It will be an imperfect estimate. We just have to be aware of the deal we have made with the devil, and interpret the results accordingly. Samples are represented in Figures 2 and 3 as being randomly selected from Figure 1.

Figure 2

Figure 3

If our sample happens to look like the three highlighted shots in Figure 2, we might go home whistling Dixie in the key of C, and certain that the rifle is shooting fine as frog’s hair split three ways. Or, we might draw the sample highlighted in Figure 3, and go home determined to work on our shooting technique, and wondering what we have done wrong. Both Figure 2 and Figure 3 are samples drawn out of the large population in Figure 1, and, as we shall see, are roughly equally likely. It is easy to be seduced into believing that Figure 2 and Figure 3 somehow represent different performance from the shooter, the rifle, or the ammunition. But we will find there is no evidence that they do.